If insects can build up 5 kilovolts while flying, then why can I zap flies with a fly-zapping tool that presumably runs at a similar or lower voltage?
Voltage is merely the "pressure" that charged particles experience. Voltage alone tells you nothing about how much charge is actually available once electricity is allowed to flow. And that's where the harm comes from. For static electricity, when you touch something, you get maybe a microcoulomb, once, and it's gone. For a power socket, you get up to 16 coulombs per second continuously.
So 3V isn't going to pose any real risk unless it's applied internally and right across a critical nerve leading to your heart or a muscle directly on the heart. For reference pacemakers are generally set to 2-3 volts. Applied externally up to ~12V is generally considered low enough voltage there's a low risk of truly adverse effects.
His entire rig at one point delivered >80k amps, but he's fine.
10 milliamps across your heart can kill but using Ohm's law we can calculate 3V / 0.010 A to get a resistance of 300 ohms. This means you're probably still going to have a bad time if you apply it directly across your heart during open-heart surgery but other than that 3 volts just isn't enough to drive a lethal current through your skin.
In the same vein, if your carpet gives you a static shock, that's likely going to be thousands of volts. But obviously there isn't actually a lot of energy stored (all you did was convert some friction), so there's next to no amperes, little work the electricity can do, and thus no harm.
When you hit a flying insect with a zapper you are supplying a high potential and low potential electrode. The insects body completes the circuit and the stored voltage is routed through the insect, rendering it a flightless blob of goo.